Modeling multi-layer flow and unsaturated flow: AEM solutions to themodified Helmholtz equation

Mark BakkerCivil Engineering, TU DelftBio and Ag Eng., Univ. of GAmarkbak@gmail.com

Sabbatical funding:TU Delft Grants Program

Flow is at steady-stateResistance to vertical flow is neglected within an aquifer

Head is function of x and y, but flow is 3DFlow in leaky layers is verticalAquifer properties are piece- wise constant

Objective: Model multi-aquifer and multi-layer flow with an analytic, mesh free approach

Funding: US EPA, WHPA, WBL, Brabant Water, Artesia, Amsterdam Water SupplyCollaborators: Otto Strack, Vic Kelson, Ken Luther

Vertical flow between aquifers or layers is computed with Darcy's law

leaky layeraquifer 1

aquifer 2

qz = k zh2−h1

H 1H 2/2=

h2−h1

c

layer 1

layer 2

qz = k zh2−h1

d=

h2−h1

c

: vertical hydraulic conductivity of leaky layerd : thickness of leaky layerc = d / : resistance

k z

k z

: vertical hydraulic conductivity of layers : thicknesses of layers

k z

c=H 1H 2/2kzH 1 , H 2

multi-aquifer flow multi-layer flow

Flow in 3-layer system is governed by 3 linked differential equations

∇2 h2 =h2−h1

c1 T 2

h2−h3

c2 T 2

∇ 2 h1 =h1−h2

c1T 1

∇ 2 h3 =h3−h2

c2 T 3

∇ 2 h=E /T

h: hydraulic head (m)

E: sink term (m/d)

T=kH: transmissivity (m2/day)

c: resistance between layers (days)

General form of deq:

System of differential equations may be written in matrix form of modified Helmholtz equation multi-layer theory of Hemker (1984)

∇ 2h1

h2

h3=

1c1 T 1

−1c1 T 1

0

−1c1T 2

1c1T 2

1c2 T 2

1c2 T 2

0 −1c2 T 3

1c2 T 3

h1

h2

h3

∇ 2h=Ah A is the system matrix and is function of aquifer properties

Hermann von Helmholtz

General form of solution for confined system with N layers: Laplace part and Helmholtz part

h=hLe∑n=1

N−1

hn un

∇ 2 hL=0

∇ 2 hn=wn hn

General form of solution:

hL fulfills Laplace's deq:

hn fulfills modified Helmholtz deq:

unit vectoreigenvector n of system matrixeigenvalue n of system matrix

eun

wn

h = Q2T tot

ln re∑n=1

N−1 An

2K0r wn unExample:

(implemented formulation is in discharge potentials)

Path lines started in top aquifer

Path lines started in bottom aquifer

A Line-sink in top aquiferDrawdown in top aquifer Drawdown bottom aquifer

An area-sink on top of the aquifer sytem, here used for recharge

Path lines started at top of aquifer

Recharge inside polygon Water mount in bottom aquifer

Multi-aquifer inhomogeneity in uniform flow

Red: k = 100 m/dBlue: k = 2 m/dGreen: leaky layer

Outside:k = 10 m/d

k = 20 m/d

Uniform flow

Heads in top aquifer

Blue: Aquifer 1Red: Aquifer 2Green: Aquifer 3

well inaq 3

well inaq 2+3

well in aq 2

recharge to top aquiferriver segments(line-sinks)

Path lines tracedback from the well

Determine sourceof well in aquifer 3

Collector well in Sonoma County, CA; model with 12 layers

Example application: modeling radial collector wells with a multi-layer approach

WHPA, Bloomington, IN

Heterogeneity in the vadose zonecauses advective spreading

Effective longitudinal spreading in saturated heterogeneous media quantified with analytic element solutions (Dagan, Fiori, Jankovic)

Analytic element solutions for unsaturated flow through heterogeneous vadose zones were developed recently (Bakker & Nieber 2004, VZJ, WRR)

(an open door)

Collaborator:John Nieber, Univ. of Minnesota

Mathematical formulation

Darcy's law for specific discharge:

qx=−k ∂∂ x

qz=−k ∂∂ z k

ψ : pressure head (negative for unsaturated flow)k(ψ) : hydraulic conductivity function

Continuity of steady flow:

x

z

∂∂ x k

∂∂ x ∂

∂ z k∂∂ z −∂k

∂ z=0

Highly n

on-line

ar deq

Kirchhoff potential and Gardner model

H =∫−∞

k sds

k =k s exp [−e]

Kirchhoff potential:

Hydraulic conductivity:(Gardner model)

ks : hydraulic conductivity at saturationα : parameter dependent on pore size distributionψe : air entry pressure head

substitution gives ....

Gustave Kirchhoff

H =k

Physically:

Kirchhoff potential is governed by linear differential equation

=H exp[−z−zc/2]

∇2=2

4

Modified Helmholtz equation

∇2 H− ∂H∂ z =0

Define:

which gives

Approach: Superimpose analytic element solutions for H(x,z), by using solutions to the Mod. Helmholtz Eq.

Hermann von Helmholtz

Analytic element solutions for circular and elliptical inhomogeneities

Hydraulic conductivity functions are different inside and outside (ks, α, and ψe may differ)

Separation of variables in radial or elliptical coordinates

Separate infinite series for inside and outside

Boundary conditions of continuity of pressure head and normal flow are met up to machine accuracy

k=k s e−e

Commonly, k of finer soil is larger than kof coarser soil under unsaturated conditions

ψe = -0.15 m

ψe = -0.25 m

Consider a finer-grained medium containing coarser grained inclusions

finer materialks = 0.36 m/dψe = -0.25 m/dα = 12.9 m-1 λ = 3.9 m-1

coarser materialks = 1 m/dψe = -0.15 m/dα = 21.2 m-1

λ = 6.2 m-1

coarser

finer

finer

uniform flow qz0

0.9 m

In saturated flow, the coarser ellipses will attract the flow

Flow patternindependentof uniform flow amount

Effect of ellipses withkcoarser ≈ 3kfineris relativelysmall

start trace experiment

compute arrival times

µ = 16.9 dσ = 0.57 dσ/µ = 0.03γ = 1.4

µ = 169 dσ = 5.7 dσ/µ = 0.03γ = 1.4

µ = 1690 dσ = 57 dσ/µ = 0.03γ = 1.4

qz0 = 1.8 cm/d

vz0 = 5.1 cm/d

qz0 = 0.18 cm/d

vz0 = 0.51 cm/d

qz0 = 0.018 cm/d

vz0 = 0.051 cm/d

Saturated flow: Breakthrough curve independent of uniform flow

qz0 = 0.05ks,fine = 1.8 cm/d

vz0 = 12.8 cm/d0.9 m

Unsaturated flow: coarse-grained ellipses divert flow

qz0 = 0.005ks,fine = 0.18 cm/d

vz0 = 2.56 cm/d

A smaller uniform flow creates a greater diversion by the ellipses

qz0 = 0.0005ks,fine = 0.018 cm/d

vz0 = 0.51 cm/d

Coarse-grained ellipses in very small uniform flow behave almost as impermeable ellipses

start trace experiment

compute arrival times

µ = 7.2 dσ = 0.99 dσ/µ = 0.14γ = 0.18

µ = 36 dσ = 8.9 dσ/µ = 0.25γ = 1.1

µ = 181 dσ = 59 dσ/µ = 0.33γ = 2.4

vz0 = 12.8 cm/d

vz0 = 2.56 cm/d

vz0 = 0.51 cm/d

Breakthrough curves for unsaturated flow

Example of flow new with many ellipses(coming soon)

Accurate and efficient models can be made of flowthrough many (thousands?) of inhomogeneities

Multi-layer solution of 3D flowto a radial collector well

Inhomogeneities in vadose zonehave much greater effect than insaturated zone and benefit fromanalytic modeling

Analytic modeling of head and flow in heterogeneous multi-layer systems and vadose zones

Mark Bakkermarkbak@gmail.com